Low Error Kramers-Kronig Estimations Using Symmetric Extrapolation Method

Abstract

Kramers-Kronig (KK) equations allow us to obtain the real or imaginary part of linear, causal and time constant functions, starting from the imaginary or real part respectively. They are normally applied on different practical applications as a control method. A common problem in measurements is the lack of data in a wide-range frequency, due to some of the inherent limitations of experiments or practical limitations of the used technology. Different solutions to this problem were proved, such as several methods for extrapolation, some of which based on piecewise polynomial fit or the approach based on the expected asymptotical behavior. In this work, we propose an approach based on the symmetric extrapolation method to generate data in missing frequency ranges, to minimize the estimated error of the KK equations. The results show that with data from impedance measurements of an electrode-electrolyte interface, the adjustment error of the transformed functions can be drastically reduced to below 1%.

Keywords: Kramers-Kronig; extrapolation; symmetric.

Fig.1

Three-electrode cell. WE: working electrode, Re1. reference electrode, CE: counter-electrode.

Fig.2

Frequency and amplitude of the interpolated points. ${\omega }_p:$central frequency of the dispersion.

Fig.3

Resistance (Black) and reactance series equivalents (Steel Blue) of the EEI versus frequency for AISI 304 stainless steel electrode polished with sandpaper #180 in NaCl 0.9% solution. Overpotential applied = 10 mV.

Fig.4

Reactance series equivalent of the ZEEI versus frequency. Experimental data (◼) and applying the KK equation (4) to the experimental data (x).

Fig.5

Resistance series equivalent of the EEI impedance. Expanded dataset (ο) and Experimental data (∙).

Fig.6

Reactance series equivalents of the EEI versus frequency. Experimental data () and applying the KK equation (4) to the expanded dataset (x).

Fig.7

Percentage change of $Z_p^{\prime \prime }\left(\Delta Z_p^{\prime \prime }\%\right)\text{\hspace{0.17em}\hspace{0.17em}versus\hspace{0.17em}}{\omega }_{min}.$Red point: case without extrapolation (figure 4). Blue point: $\Delta Z_p^{\prime \prime }\%=-1.$